模糊随机变量序列的极限定理
|
摘要
处理数据时，人们总是将不确定性与随机性联系在一起。实际上，由于测量的主观性与人类知识和理解的不精确性，数据本身存在着区别于随机性的不确定因素，称之为模糊性。人们希望在使用传统的统计理论同时，也可以将数据的模糊性考虑进去，因此模糊集被用来表示数据，取代了传统的实数或多元实数，模糊数和模糊随机变量的概念应运而生。在众多学者的努力下，近二十多年来，模糊随机理论得到了巨大的发展。模糊随机变量作为传统的实值随机变量概念的延伸，其性质已经受到广泛的研究。
类似于实值概率理论和经典的数理统计理论，模糊随机变量序列的极限定理是模糊随机理论的重要组成部分，也是模糊统计分析的重要理论基础。实值随机变量取值于实数空间R，模糊随机变量取值于模糊数空间E~d。然而，E~d较尺要复杂得多。
E~d上的距离有多种，在不同距离定义下，模糊数与模糊随机变量序列的收敛性会有所不同。在一致Hausdorff距离D下，相关研究已比较深入，而针对图距离和一种D_2~*距离下关于模糊随机变量序列的收敛性研究不是很丰富。本文的工作围绕关于这两种距离的收敛性展开。
在图距离下，本文给出了模糊随机变量序列几乎处处收敛的判别定理。Ogura和Li曾得到一个判别条件，但有局限之处。本文对该判别条件进行了改进，将原来随机集序列的收敛条件改进为实值随机变量序列的按点收敛。该定理在模糊随机理论与实值概率论之间建立了桥梁。利用改进的判别定理，本文将实值概率论中的控制收敛定理和部分Marcinkiewicz-Zygmund型强大数定律推广到了模糊随机情形。Kolmogorov型强大数定律为Marcinkiewicz-Zygmund型的一个推论。
实值概率论中，独立随机变量和的性质是核心内容之一。其中，Kolmogorov三级数定理又是非常重要的定理。D_2~*距离下，本文进行了模糊随机理论中类似命题的研究。由于模糊随机变量的方差定义是利用可分距离D_2~*，因此在这个距离下研究独立模糊随机变量和的收敛性显得非常自然。本文首先利用颜云志等的方法得到完备可分的距离空间((?)_2~d，D_2~*)，然后通过探究取值于该空间的模糊随机变量序列与模糊随机变量级数的均方收敛与几乎处处收敛、依概率收敛之间的关系，得到推广了的Kolmogorov三级数定理。证明过程中，距离空间((?)_2~d，D_2~*)的良好性质起了很大作用。
When dealing with data, uncertainty is mainly incorporated in the sense of randomness. But it is reasonable to suppose that data also inherit uncertainty in the sense of vagueness, due to subjective judgement, imprecise human knowledge and perception. In order to integrate vagueness into techniques of statistical inference, recent developments during the last two decades propose to represent data by fuzzy subsets rather than real numbers or tuples of real numbers. Extending the basic notion of random variables, the properties of fuzzy random variables has been investigated.
The extension of mathematical statistics to vague data may be regarded as reasonable and well founded if analogues of classical limit theorems like the strong law of large numbers and the convergence theorems of series are formulated. The classical limit theorems base, simultaneously, on a fixed notion of random variable according to a fixed metric space R. Some convergence properties are obtained by the specific characteristics of this metric space.
The aim of this paper is to give analogues of the classical limit theorems which related to some different concepts of convergence and based on different metric spaces. In the uniform metric D, related subject has been widely discussed. Seldom results appear, however, with respect to sendograph and D_2~* metric. This paper mainly deals with the latter two metrics.
In sendograph metric, a criterion of almost sure convergence for fuzzy random variables in sendograph metric is established. Ogura and Li have obtained another criterion which leads to a similar result. They show that the sendograph convergence follows from the convergence of the sequences of the level sets for rational-number levels, which are random sets, nevertheless will cause some limitations. Our criterion is an improved version, which turns that condition into the pointwise convergence of the real-valued random variables. As an application, one dominated convergence theorem in standard probability theory is extended to the case of fuzzy random variables. A part of Marcinkiewicz-Zygmund's type strong law of large numbers - and Kolmogorov's type, as a trivial corollary - is proved in the field of fuzzy random theory using the same technique.
One paramount concern in standard probability theory is the behavior of series, i.e. sums of independent random variables. Kolmogorov three series theorem is of particularly importance. This paper deals with an ana- logue of it with respect to D_2~* metric. A variance series of independent random variables is included in the classical theorem. Under our circumstance, the definition of variance is based on the separable metric D_2~*, hence D_2~* is more reasonable than the metric D in the discussion of 2nd-order moment of fuzzy random variables in reference to Feng. In this paper, we first obtain a separable and complete metric space (E|^_2~d, D_2~*) by modifying the definition of fuzzy numbers using Yan's method. Next, the relationships among different types of convergence like mean-square convergence, almost sure convergence and convergence in probability are investigated. Our version of Kolmogorov three series theorem is finally proved by applying the above relationships and the properties of (E_^_2~d, D_2~*).
类似于实值概率理论和经典的数理统计理论，模糊随机变量序列的极限定理是模糊随机理论的重要组成部分，也是模糊统计分析的重要理论基础。实值随机变量取值于实数空间R，模糊随机变量取值于模糊数空间E~d。然而，E~d较尺要复杂得多。
E~d上的距离有多种，在不同距离定义下，模糊数与模糊随机变量序列的收敛性会有所不同。在一致Hausdorff距离D下，相关研究已比较深入，而针对图距离和一种D_2~*距离下关于模糊随机变量序列的收敛性研究不是很丰富。本文的工作围绕关于这两种距离的收敛性展开。
在图距离下，本文给出了模糊随机变量序列几乎处处收敛的判别定理。Ogura和Li曾得到一个判别条件，但有局限之处。本文对该判别条件进行了改进，将原来随机集序列的收敛条件改进为实值随机变量序列的按点收敛。该定理在模糊随机理论与实值概率论之间建立了桥梁。利用改进的判别定理，本文将实值概率论中的控制收敛定理和部分Marcinkiewicz-Zygmund型强大数定律推广到了模糊随机情形。Kolmogorov型强大数定律为Marcinkiewicz-Zygmund型的一个推论。
实值概率论中，独立随机变量和的性质是核心内容之一。其中，Kolmogorov三级数定理又是非常重要的定理。D_2~*距离下，本文进行了模糊随机理论中类似命题的研究。由于模糊随机变量的方差定义是利用可分距离D_2~*，因此在这个距离下研究独立模糊随机变量和的收敛性显得非常自然。本文首先利用颜云志等的方法得到完备可分的距离空间((?)_2~d，D_2~*)，然后通过探究取值于该空间的模糊随机变量序列与模糊随机变量级数的均方收敛与几乎处处收敛、依概率收敛之间的关系，得到推广了的Kolmogorov三级数定理。证明过程中，距离空间((?)_2~d，D_2~*)的良好性质起了很大作用。
When dealing with data, uncertainty is mainly incorporated in the sense of randomness. But it is reasonable to suppose that data also inherit uncertainty in the sense of vagueness, due to subjective judgement, imprecise human knowledge and perception. In order to integrate vagueness into techniques of statistical inference, recent developments during the last two decades propose to represent data by fuzzy subsets rather than real numbers or tuples of real numbers. Extending the basic notion of random variables, the properties of fuzzy random variables has been investigated.
The extension of mathematical statistics to vague data may be regarded as reasonable and well founded if analogues of classical limit theorems like the strong law of large numbers and the convergence theorems of series are formulated. The classical limit theorems base, simultaneously, on a fixed notion of random variable according to a fixed metric space R. Some convergence properties are obtained by the specific characteristics of this metric space.
The aim of this paper is to give analogues of the classical limit theorems which related to some different concepts of convergence and based on different metric spaces. In the uniform metric D, related subject has been widely discussed. Seldom results appear, however, with respect to sendograph and D_2~* metric. This paper mainly deals with the latter two metrics.
In sendograph metric, a criterion of almost sure convergence for fuzzy random variables in sendograph metric is established. Ogura and Li have obtained another criterion which leads to a similar result. They show that the sendograph convergence follows from the convergence of the sequences of the level sets for rational-number levels, which are random sets, nevertheless will cause some limitations. Our criterion is an improved version, which turns that condition into the pointwise convergence of the real-valued random variables. As an application, one dominated convergence theorem in standard probability theory is extended to the case of fuzzy random variables. A part of Marcinkiewicz-Zygmund's type strong law of large numbers - and Kolmogorov's type, as a trivial corollary - is proved in the field of fuzzy random theory using the same technique.
One paramount concern in standard probability theory is the behavior of series, i.e. sums of independent random variables. Kolmogorov three series theorem is of particularly importance. This paper deals with an ana- logue of it with respect to D_2~* metric. A variance series of independent random variables is included in the classical theorem. Under our circumstance, the definition of variance is based on the separable metric D_2~*, hence D_2~* is more reasonable than the metric D in the discussion of 2nd-order moment of fuzzy random variables in reference to Feng. In this paper, we first obtain a separable and complete metric space (E|^_2~d, D_2~*) by modifying the definition of fuzzy numbers using Yan's method. Next, the relationships among different types of convergence like mean-square convergence, almost sure convergence and convergence in probability are investigated. Our version of Kolmogorov three series theorem is finally proved by applying the above relationships and the properties of (E_^_2~d, D_2~*).
引文
[1] R. J. Aumann. Integrals of set-valued functions. Fuzzy Sets and Systems, 12: 1-12, 1965.
[2] J. J. Buckley and A. Yan. Fuzzy topological vector spaces over (?). Fuzzy Sets and Systems, 105: 259-275, 1999.
[3] J. J. Buckley and A. Yan. Fuzzy functional analysis (Ⅰ): Basic concepts. Fuzzy Sets and Systems, 115: 393-402, 2000.
[4] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 35:241-249, 1990.
[5] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets: Corrigendum. Fuzzy Sets and Systems, 45:123, 1992.
[6] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets-Theory and Applications. World Scientific, Singapore, 1994.
[7] P. Diamond, P. Kloeden, and A. Pokrovskii. Absolute retracts and a general fixed point theorem for fuzzy sets. Fuzzy Sets and Systems, 86:377-380, 1997.
[8] T. Fan. On the compactness of fuzzy numbers with sendograph metric. Fuzzy Sets and Systems, 143: 471-477, 2004.
[9] T. Fan and G. Wang. Endographic approach on supremum and infimum of fuzzy numbers. Information Sciences, 159:221-231, 2004.
[10] Y. Feng. Convergence theorems for fuzzy random variables and fuzzy martingales. Fuzzy Sets and Systems, 103: 435-441, 1999.
[11] Y. Feng. Mean-square integral and differential of fuzzy stochastic processes. Fuzzy Sets and Systems, 102: 271-280, 1999.
[12] Y. Feng. Fuzzy-valued mappings with finite variation, fuzzy-valued measures and fuzzy-valued Lebesgue-Stieltjes integrals. Fuzzy Sets and Systems, 121: 227-236, 2001.
[13] Y. Feng. Sums of independent fuzzy random variables. Fuzzy Sets and Systems, 123:11-18, 2001.
[14] Y. Feng. Approach to generalize laws of large numbers for fuzzy random variables. Fuzzy Sets and Systems, 128: 237-245, 2002.
[15] Y. Feng. On the convergence of fuzzy set sequences. Journal of Donghua University (Eng. Ed.), 4(19): 29-32, 2002.
[16] Y. Feng. Metrics on noncompact fuzzy number space E~n. Journal of Donghua University (Eng. Ed.), 21(2): 83-87, 2004.
[17] Y. Feng. Note on "sums of independent fuzzy random variables". Fuzzy Sets and Systems, 143: 479-485, 2004.
[18] Y. Feng. Strong law of large numbers for stationary sequences of random upper semicontinuous functions. Stoch. Anal. Appl., 22: 1067-1083, 2004.
[19] Y. Feng, L. Hu, and H. Shu. The variance and covariance of fuzzy random variables and their applications. Fuzzy Sets and Systems, 120(3): 487-497, 2001.
[20] G. H. Greco. A characterization of relatively compact sets of fuzzy sets. Nonlinear Analysis, 64: 518-529, 2006.
[21] G. H. Greco. Sendograph metric and relatively compact sets of fuzzy sets. Fuzzy Sets and Systems, 157: 286-291, 2006.
[22] G. H. Greco, M. P. Moschen, and E. Quelho Frota Rezende. On the variational convergence for fuzzy sets in metric spaces. Ann. Univ. Ferrara Sez. Ⅶ Sc. Mat., 44: 27-39, 1998.
[23] O. Kaleva. On the equivalence of fuzzy sets. Fuzzy Sets and Systems, 17: 53-65, 1985.
[24] D. S. Kim and Y. K. Kim. Some properties of a new metric on the space of fuzzy numbers. Fuzzy Sets and Systems, 145: 395-410, 2004.
[25] E. P. Klement, M. L. Puri, and D. A. Ralescu. Limit theorems for fuzzy random variables. Proceedings of the Royal Society of London. Series A, Mathematatical and Physical Sciences, 407(1832): 171-182, 1986.
[26] P. Kloeden. Compact supported endographs and fuzzy sets. Fuzzy Sets and Systems, 4: 192-201, 1980.
[27] P. Kloeden. Fuzzy dynamical systems. Fuzzy Sets and Systems, 7: 275-296, 1982.
[28] R. Korner. On the variance of fuzzy random variables. Fuzzy Sets and Systems, 92: 83-93, 1997.
[29] V. Kratschmer. A unified approach to fuzzy random variables. Fuzzy Sets and Systems, 123: 1-9, 2001.
[30] V. Kratschmer. Limit theorems for fuzzy-random variables. Fuzzy Sets and Systems, 126: 253-263, 2002.
[31] H. Kwakernaak. Fuzzy random variables-I. Definitions and theorems. Information Sciences, 15: 1-29, 1978.
[32] S. Li and Y. Ogura. Convergence in graph for fuzzy valued martingales and smartingales. Statistical Modeling, Analysis and Managenment of Fuzzy Data, 87:67-85, 2002.
[33] M. Loeve. Probability Theory I, 4th ed. Springer-Verlag, New York, 1977.
[34] M. A. Lubiano, M. A. Gil, M. Lopez-Diaz, and M.T. L6pez. The (?)-mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets and Systems, 111: 307-317, 2000.
[35] W. Nather. Random fuzzy variables of second order and applications to statistical inference. Information Sciences, 133:69-88, 2001.
[36] Y. Ogura and S. Li. Separability for graph convergence of sequences of fuzzy-valued random variables. Fuzzy Sets and Systems, 123: 19-27,2001.
[37] M. L. Puri and D. A. Ralescu. Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114: 151-158, 1986.
[38] M. V. K. Ravi Kishore and M. Parvaiz. On supported endographs and fuzzy sets. Fuzzy Sets and Systems, 148: 329-332, 2004.
[39] M. Rojas-Medar and H. Roman-Flores. On the equivalence of convergences of fuzzy sets. Fuzzy Sets and Systems, 80:217-224, 1996.
[40] H. -C. Wu. The fuzzy estimators of fuzzy parameters based on fuzzy random variables. European Journal of Operational Research, 146: 101-114, 2003.
[41] L. A. Zadeh. Fuzzy sets. Information and Control, 8:338-353, 1965.
[42] 冯德兴．凸分析基础．科学出版社，北京，1995．
[43] 刘军，冯玉瑚．模糊数序列的收敛性．东华大学学报，(已录用)，2006．
[44] 颜云志，冯玉瑚．非紧模糊数的广义距离空间．东华大学学报，30(4)：21-25，2004．
[45] 张文修．集值测度与随机集．西安交通大学出版社，西安，1989．
[2] J. J. Buckley and A. Yan. Fuzzy topological vector spaces over (?). Fuzzy Sets and Systems, 105: 259-275, 1999.
[3] J. J. Buckley and A. Yan. Fuzzy functional analysis (Ⅰ): Basic concepts. Fuzzy Sets and Systems, 115: 393-402, 2000.
[4] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 35:241-249, 1990.
[5] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets: Corrigendum. Fuzzy Sets and Systems, 45:123, 1992.
[6] P. Diamond and P. Kloeden. Metric spaces of fuzzy sets-Theory and Applications. World Scientific, Singapore, 1994.
[7] P. Diamond, P. Kloeden, and A. Pokrovskii. Absolute retracts and a general fixed point theorem for fuzzy sets. Fuzzy Sets and Systems, 86:377-380, 1997.
[8] T. Fan. On the compactness of fuzzy numbers with sendograph metric. Fuzzy Sets and Systems, 143: 471-477, 2004.
[9] T. Fan and G. Wang. Endographic approach on supremum and infimum of fuzzy numbers. Information Sciences, 159:221-231, 2004.
[10] Y. Feng. Convergence theorems for fuzzy random variables and fuzzy martingales. Fuzzy Sets and Systems, 103: 435-441, 1999.
[11] Y. Feng. Mean-square integral and differential of fuzzy stochastic processes. Fuzzy Sets and Systems, 102: 271-280, 1999.
[12] Y. Feng. Fuzzy-valued mappings with finite variation, fuzzy-valued measures and fuzzy-valued Lebesgue-Stieltjes integrals. Fuzzy Sets and Systems, 121: 227-236, 2001.
[13] Y. Feng. Sums of independent fuzzy random variables. Fuzzy Sets and Systems, 123:11-18, 2001.
[14] Y. Feng. Approach to generalize laws of large numbers for fuzzy random variables. Fuzzy Sets and Systems, 128: 237-245, 2002.
[15] Y. Feng. On the convergence of fuzzy set sequences. Journal of Donghua University (Eng. Ed.), 4(19): 29-32, 2002.
[16] Y. Feng. Metrics on noncompact fuzzy number space E~n. Journal of Donghua University (Eng. Ed.), 21(2): 83-87, 2004.
[17] Y. Feng. Note on "sums of independent fuzzy random variables". Fuzzy Sets and Systems, 143: 479-485, 2004.
[18] Y. Feng. Strong law of large numbers for stationary sequences of random upper semicontinuous functions. Stoch. Anal. Appl., 22: 1067-1083, 2004.
[19] Y. Feng, L. Hu, and H. Shu. The variance and covariance of fuzzy random variables and their applications. Fuzzy Sets and Systems, 120(3): 487-497, 2001.
[20] G. H. Greco. A characterization of relatively compact sets of fuzzy sets. Nonlinear Analysis, 64: 518-529, 2006.
[21] G. H. Greco. Sendograph metric and relatively compact sets of fuzzy sets. Fuzzy Sets and Systems, 157: 286-291, 2006.
[22] G. H. Greco, M. P. Moschen, and E. Quelho Frota Rezende. On the variational convergence for fuzzy sets in metric spaces. Ann. Univ. Ferrara Sez. Ⅶ Sc. Mat., 44: 27-39, 1998.
[23] O. Kaleva. On the equivalence of fuzzy sets. Fuzzy Sets and Systems, 17: 53-65, 1985.
[24] D. S. Kim and Y. K. Kim. Some properties of a new metric on the space of fuzzy numbers. Fuzzy Sets and Systems, 145: 395-410, 2004.
[25] E. P. Klement, M. L. Puri, and D. A. Ralescu. Limit theorems for fuzzy random variables. Proceedings of the Royal Society of London. Series A, Mathematatical and Physical Sciences, 407(1832): 171-182, 1986.
[26] P. Kloeden. Compact supported endographs and fuzzy sets. Fuzzy Sets and Systems, 4: 192-201, 1980.
[27] P. Kloeden. Fuzzy dynamical systems. Fuzzy Sets and Systems, 7: 275-296, 1982.
[28] R. Korner. On the variance of fuzzy random variables. Fuzzy Sets and Systems, 92: 83-93, 1997.
[29] V. Kratschmer. A unified approach to fuzzy random variables. Fuzzy Sets and Systems, 123: 1-9, 2001.
[30] V. Kratschmer. Limit theorems for fuzzy-random variables. Fuzzy Sets and Systems, 126: 253-263, 2002.
[31] H. Kwakernaak. Fuzzy random variables-I. Definitions and theorems. Information Sciences, 15: 1-29, 1978.
[32] S. Li and Y. Ogura. Convergence in graph for fuzzy valued martingales and smartingales. Statistical Modeling, Analysis and Managenment of Fuzzy Data, 87:67-85, 2002.
[33] M. Loeve. Probability Theory I, 4th ed. Springer-Verlag, New York, 1977.
[34] M. A. Lubiano, M. A. Gil, M. Lopez-Diaz, and M.T. L6pez. The (?)-mean squared dispersion associated with a fuzzy random variable. Fuzzy Sets and Systems, 111: 307-317, 2000.
[35] W. Nather. Random fuzzy variables of second order and applications to statistical inference. Information Sciences, 133:69-88, 2001.
[36] Y. Ogura and S. Li. Separability for graph convergence of sequences of fuzzy-valued random variables. Fuzzy Sets and Systems, 123: 19-27,2001.
[37] M. L. Puri and D. A. Ralescu. Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114: 151-158, 1986.
[38] M. V. K. Ravi Kishore and M. Parvaiz. On supported endographs and fuzzy sets. Fuzzy Sets and Systems, 148: 329-332, 2004.
[39] M. Rojas-Medar and H. Roman-Flores. On the equivalence of convergences of fuzzy sets. Fuzzy Sets and Systems, 80:217-224, 1996.
[40] H. -C. Wu. The fuzzy estimators of fuzzy parameters based on fuzzy random variables. European Journal of Operational Research, 146: 101-114, 2003.
[41] L. A. Zadeh. Fuzzy sets. Information and Control, 8:338-353, 1965.
[42] 冯德兴．凸分析基础．科学出版社，北京，1995．
[43] 刘军，冯玉瑚．模糊数序列的收敛性．东华大学学报，(已录用)，2006．
[44] 颜云志，冯玉瑚．非紧模糊数的广义距离空间．东华大学学报，30(4)：21-25，2004．
[45] 张文修．集值测度与随机集．西安交通大学出版社，西安，1989．